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TitleInflow Performance Relationship Wiggins, M.L
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SPE 25458

Generalized Inflow Performance Relationships for
Three-Phase Flow

Society of Petroleum Engineers

M.L. Wiggins, U. of Oklahoma

SPE Member

Copyright 1993, Society of Petroleum Engineers, Inc.

This paper was prepared for presentation at the Production Operations Symposium held in Oklahoma City, OK, U.S.A., March 21-23, 1993.

This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the author(s). Contents of the paper,
as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the author(s). The material, as presented, does not necessarily reflect
any position of the Society of Petroleum Engineers, its officers, or members. Papers presented at SPE meetings are subject to publication review by Editorial Committees of the Society
of Petroleum Engineers. Permission to copy is restricted to an abstract of not more than 300 words. Illustrations may not be copied. The abstract should contain conspicuous acknowledgment
of where and by whom the paper is presented. Write Librarian, SPE, P.O. Box 833836, Richardson, TX 75083·3836, U.S.A. Telex, 163245 SPEUT.

ABSTRACT

Generalized three-phase inflow
performance relationships (IPRs) for the oil
and water phases are presented in this paper.
These relationships yield adequate estimates of
the production-pressure behavior of oil wells
producing from homogeneous, bounded
reservoirs during boundary-dominated flow.
The IPRs are empirical relationships based on
linear regression analysis of simulator results
and cover a wide range of reservoir fluid and
rock properties. Methods to study the effects of
changes in flow efficiency and to predict future
performance are also presented.

INTRODUCTION

Predicting the performance of
individual oil wells is an important
responsibility of the petroleum engineer.
Reasonable estimates of well performance allow
the engineer to determine the optimum
production scheme, design production and
artificial lift equipment, design stimulation
treatments and forecast production for
planning purposes. Each of these items is
important to the efficient operation of
producing wells and successful reservoir
management.

When estimating oil well performance,
it is often assumed that fluid inflow is
proportional to the difference between reservoir
pressure and wellbore pressure. One of the
first relationships to be used based on this
assumption was the Productivity Index (PI).
This straight-line relationship can be derived
from Darcy'sllaw for the steady-state flow of a

single incompressible fluid and is the ratio of
the producing rate to the pressure difference.
However, Evinger and Muskat2,3 pointed out
that a straight-line relationship should not be
expected when multiple phases are flowing in
the reservoir. They presented theoretical
calculations that showed a curved relationship
between flow rate and pressure for two- and
three-phase flow.

Vogel4 later developed an empirical
inflow performance relationship (IPR) for
solution-gas drive reservoirs that accounted for
the flow of two phases, oil and gas, in the
reservoir based on computer simulation results.
The resulting IPR equation is

483

~ = 1 - 0.2 Pwf - 0.8 (Pwfj2
qo;nax Pr Pr

(1)

Fetkovich5 also presented an empirical inflow
performance relationship based on field data
that has gained wide acceptance. His
relationship, of a form similar to the empirical
gas well deliverability equation proposed by
Rawlins and Schellhardt6 , is

(2)

Both Vogel's and Fetkovich's relations were
developed for solution-gas drive reservoirs and
are widely used due to their simplicity.

In an attempt to extend Vogel's
approach to three-phase flow, Brown7

presented a method proposed by Petrobras for
determining the inflow performance of oil wells

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2 GENERALIZED INFLOW PERFORMANCE RELATIONSHIPS
FOR THREE-PHASE FLOW

SPE 25458

producing water. The method uses a constant
PI for the water production and adds it to a
Vogel relation for the oil production to obtain a
composite inflow performance relationship.
Sukarno8 proposed a method derived from
computer simulation of three-phase flow. This
method resulted from nonlinear regression
analysis of the generated simulator results and
is based on the producing water cut and total
liquid flow rate. The resulting relationship is
a quadratic whose coefficients are functions of
water cut. As of yet, no one has addressed the
problem of predicting future performance or
studied the effect of a skin region around the
wellbore during three-phase flow.

In this paper, generalized inflow
performance relationships are presented for
three-phase flow in bounded, homogeneous
reservoirs. The proposed IPRs are compared
with other three-phase methods currently
available. The methods presented are based on
homogeneous reservoirs where gravity and
capillary effects are negligible. Methods are
also presented for predicting performance when
reservoir conditions change from test
conditions. This includes predicting future
performance due to depletion and predicting
performance when changes occur in the skin
region near the wellbore.

GENERALIZED IPRs

Wiggins, Russell and Jennings9

recently proposed an analytical IPR for three-
phase flow in bounded reservoirs. An
advantage of the analytical IPR is that one can
develop an IPR specific to a particular
reservoir and its operating conditions. The
major disadvantage, however, is that it
requires knowledge of relative permeability
and reservoir fluid properties and how they
behave with pressure. This is not a large
obstacle if relative permeability and pressure-
volume-temperature data are available for the
reservoir of interest, along with an idea of the
average reservoir pressure and water
saturation. With this information, one can
develop the required mobility function profiles
from the current reservoir pressure to near-

484

zero flowing pressure. This profile can then be
used to develop the analytical IPRs for the oil
and water phases.

Unfortunately, we do not always have
reliable relative permeability or fluid property
information. In this case, the analytical IPR is
only of academic interest in our operations. To
overcome this problem, generalized three-phase
IPRs similar to Vogel's were developed and are
presented here. The resulting IPR equations
are based on regression analysis of simulator
results covering a wide range of relative
permeability information, fluid property data
and water saturations.

Development of Simulator Results

To develop the generalized equations to
predict inflow performance, IPR curves were
generated from simulator results for four basic
sets of relative permeability and fluid property
data. Each set of data was used to generate
simulator results from irreducible water
saturation to residual oil saturation. Sixteen
theoretical reservoirs were examined from
initial pressure to the minimum flowing
bottomhole pressure. Table 1 presents the
range of reservoir properties used in the
development of the generalized IPRs.

Simulator results were obtained for a
radial flow geometry and constant oil rate
production. Maximum oil and water
production rates were estimated at each stage
of depletion from the simulator results at a
minimum flowing bottomhole pressure of 14.7
psia. If the flowing bottomhole pressure did
not reach this minimum during the simulation,
the maximum rate was estimated from the
production information available and then
checked by rerunning the simulator.

Figs. 1 and 2 present typical oil and
water inflow performance curves for Case 3
with an initial water saturation of 20% at
several stages of depletion. These curves have
the same characteristic concave shape noticed
by Vogel in his research. The curves were
normalized by dividing each point of

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6 GENERALIZED INFLOW PERFORMANCE RELATIONSHIPS
FOR THREE-PHASE FLOW

SPE 25458

method. The results presented in these tables
indicate that the error increases as we estimate
further in time, however, on an absolute basis,
the predictions are within reasonable
engineering accuracy.

The analysis suggests that care should
be taken in estimating future performance over
large stages of depletion as the error may
increase. This error may not be significant if
the absolute difference in production values are
small, as indicated by several of the examples.
Based on analysis of information used in
developing this method, one should exercise
caution in predicting future rates at reservoir
pressure ratios less than 70%. While estimates
at pressure ratios less than 70% may be
relatively accurate, they may contain
significant errors. It is recommended that
initial future performance estimates be
updated every six months to one year. This
would progressively reduce the uncertainty in
earlier estimates as depletion occurs in the
reservoir.

APPLICABILITY

The proposed IPRs and methods
presented in this research for three-phase flow
were developed from analysis of multiphase
flow in bounded, homogeneous reservoirs where
there is no external influx of fluids into the
reservoir, and apply to the boundary-
dominated flow regime. The methods are
limited by the following assumptions: 1) all
reservoirs are initially at the bubble point; 2)
no initial free gas phase is present; 3) a mobile
water phase is present for three-phase studies;
4) Darcy's law for multiphase flow applies; 5)
isothermal conditions exist; 6) there is no
reaction between reservoir fluids and reservoir
rock; 7) no gas solubility exists in the water; 8)
gravity effects are negligible; and 9) there is a
fully penetrating wellbore. Strictly speaking,
the methods cannot be considered correct when
other types of reservoir conditions exist, and
the engineer should exercise great care in
utilizing the proposed methods.

From a practical viewpoint the

488

proposed methods may have limited
applicability, since very few reservoirs
completely satisfy the assumptions. One might
speculate that the methods have merit under
less stringent conditions than those under
which they were developed. Examples would
include: reservoirs that have very limited water
influx; reservoirs that initially had no mobile
water phase but began producing water due to
limited water influx; large reservoirs
experiencing water influx where portions of the
reservoir are isolated from the influx by
producing wells nearer the reservoir
boundaries. Other examples might include
reservoirs that are relatively thin with respect
to the drainage area where gravity effects are
negligible, and partially penetrating wells
where there is little vertical permeability.
These examples are only speculation and
further research is required before the
proposed methods can be extended to these
situations.

CONCLUSIONS

1. Generalized three-phase IPRs have
been presented that are suitable for use over a
wide range of reservoir properties. The
proposed relationships are Vogel-type IPRs
that require single point estimates of oil and
water production rates, flowing wellbore
pressure and average reservoir pressure.

2. The generalized IPRs have been
verified using information presented by
Sukarno and by comparison to the three-phase
methods of Brown and Sukarno. The proposed
method yielded results as reliable as these two
methods while being much simpler to use.

3. A method has been presented to
estimate pressure-production behavior due to
changes in flow efficiency. The method
appears to yield suitable results with
maximum errors between the predictions and
simulator results being less than 15% for the
cases studied. This error includes errors from
the generalized IPR and the definition of flow
efficiency.

Page 7

SPE 25458 MICHAEL L. WIGGINS 7

4. A method has been proposed for
predicting future performance that is similar in
form to a Vogel-type IPR The method is
suggested by the Taylor series expansion of the
multiphase flow equations proposed by
Wiggins, Russell and Jennings. To the
author's knowledge, no one has proposed a
method for predicting future performance
during three-phase boundary-dominated flow.

NOMENCLATURE

Ef
~
p

Pr
Pwf

CIo
CIo,rnax

oil formation volume factor,
RB/STB
flow efficiency, dimensionless
relative permeability to oil
pressure, psi
average reservoir pressure, psi
flowing wellbore pressure, psi
oil production rate, BOPD
maximum oil production rate,
BOPD
water production rate, BWPD
maximum water production
rate, BWPD
external boundary radius, ft
wellbore radius, ft
skin factor, dimensionless
regression coefficient
oil viscosity, cp

REFERENCES

1. Darcy, H.: Les Fontaines Publiques de
la Ville de Dijon, Victor Dalmont, Paris
(1856) 590-594.

2. Evinger, H.H. and Muskat, M.:
"Calculation of Theoretical Productivity
Factors", Trans., AIME (1942) 146,
126-139.

3.

4.

5.

6.

7.

8.

9.

10.

489

Evinger, H.H. and Muskat, M.:
"Calculation of Productivity Factors for
Oil-gas-water Systems in the Steady
State", Trans., AIME (1942) 146, 194-
203.
Vogel, J.V.: "Inflow Performance
Relationships for Solution-Gas Drive
Wells", JPT (Jan. 1968) 83-92.
Fetkovich, M.J.: "The Isochronal
Testing of Oil Wells", paper SPE 4529
presented at the 1973 SPE Annual
Meeting, Las Vegas, NV, Sept. 30 - Oct.
3.
Rawlins, E.L. and Schellhardt, M.A:
Backpressure Data on Natural Gas
Wells and Their Application to
Production Practices, USBM (1935) 7.
Brown, KE.: The Technology of
Artificial Lift Methods, PennWell
Publishing Co., Tulsa, OK (1984) 4, 18-
35.
Sukarno, P.: "Inflow Performance
Relationship Curves in Two-Phase and
Three-Phase Flow Conditions", Ph.D.
dissertation, U. of Tulsa, Tulsa, OK
(1986).
Wiggins, M.L., Russell, J.E. and
Jennings, J.W.: "Analytical Inflow
Performance Relationships for Three-
Phase Flow in Bounded Reservoirs",
paper SPE 24055 presented at the
1992 Western Regional Meeting,
Bakersfield, CA, Mar. 30-Apr. 1.
Freund, RJ. and Littell, RC.: SAS
System for Regression, SAS Institute,
Cary, NC (1986).

Page 11

SPE 25458 MICHAEL L. WIGGINS 11

1000 2000

pwf,psia

3000 4000

Fig. 1. Oil inflow performance curves for Case 3, 20% Swi, at
several stages of depletion generated from simulator results.

1.0
0

0

%
0

0.8 00

cDo
/1

~
0.6 'b

e DC
g.

t5% ...... 0
0" 0.4 0

00
0

&
0.2

,
\

0.0
0.0 0.2 0.4 0.6 0.8 1.0

pwf/pr

Fig. 3. OilIPR curves for Case 3, 20% Swi.

493

Fig. 2 Water inflow performance curves for Case 3, 20% Swi,
at several stages of depletion generated from simulator results.

1.0
0

0

°cP
0.8

0

0
00

0

l;l 0.6 0

~
a:t

0 ......
~ [JJ
0" 0.4 Bo

0

Cu::J
0.2

IC
QJ

c

~
0.0

0.0 0.2 0.4 0.6 0.8 1.0

pwf/pr

Fig. 4. Water IPR curves for Case 3, 20% Swi.

Page 12

12 GENERALIZED INFLOW PERFORMANCE RELATIONSIDPS
FOR THREE-PHASE FLOW

SPE 25458

1;l

~
"-
S-

1.0
• Simulator Results

0.8

0.6

0.4

0.2

qo/qomax -1.0000 - 0519167 (Pwf/pr) - 0.481092 (pwf/prl"2
0.0

0.0 0.2 0.4 0.6 0.8

pwf/pr

1.0

Fig. 5. Comparison of simulator results with generalized oil IPR.

1.0

• Simulator Results
0.8

- Pmposed Relation

y - 0.15376309 x + D.83516299 ><"'2

Po. x
e 0.6
g.
"-
] • Ii.
~ 0.4 'II'"
C7' .......

. F'alf. • •

.. .
0.2

.. . ..
O.o~~~-'----~-----r----~ __ r-~-----r--------i

0.0 0.2 0.4 0.6 0.8

pr,f/pr,p

Fig. 7. Comparison of simulator results to proposed method
for determining future performance for the oil phase.

1.0

494

Po.
1;l'
Ii.
~
0'
"-.....
1;l

l

1.0

0.8

1;l 0.6

t
"-

~
0.4

0.2

• Simulator Results

qw / qwmax - 1.0000 - 0.722235 (pwf/pr)
-0.284m (pwf/prl"2

O'O+-----~_r----~----r-~----,_----~_,----~~
0.0 0.2 0.4 0.6 0.8 1.0

pwf/pr

Fig. 6. Comparison of simulator results with generalized water IPR.

1.0

• Simulator Results
- Proposed Relation

0.8

Y - 059245433x + 0.364?9178x"2

0.6

0.4

0.2

o.o~~~-r----~~-----------'------~-r------~
0.0 0.2 0.4 0.6 0.8 1.0

pr,f/pr,p

Fig. 8. Comparison of simulator results to proposed method
for determining future performance for the water phase.

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